Multi-Objective Optimization

Multi-task learning (MTL) requires navigating unavoidable trade-offs among competing objectives. This paradigm is frequently formulated as multi-objective optimization (MOO), where the scalarization is favored to reduce an MOO problem to a single objective. We empirically find that existing merit-function-based scalarization approaches are sensitive to the relative scales of different objectives in practical MTL, where task losses commonly differ by orders of magnitude. The optimization process often favors objectives with larger scales even though the underlying Pareto optimal solutions remains invariant to rescaling (i.e., multiplying an objective by a positive constant). To address this issue, we propose Scale-Invariant Merit-function-based Scalarization (SIMS) for MTL. Specifically, SIMS adopts a transformation-induced merit function to convert the MOO problem of MTL to a single objective that renders optimization invariant to the magnitudes of losses. Theoretically, we prove that the requirement for scale invariance uniquely determines this transformation to be logarithmic. We further show that this general transformation-induced merit function preserves weak Pareto optimality and admits a smooth surrogate with controllable approximation error. Extensive experiments on representative multi-task benchmarks demonstrate that SIMS consistently outperforms existing scalarization methods and achieves state-of-the-art performance.

Applying knowledge transfer across multiple optimization tasks, multitask optimization (MTO) emerges as a promising approach to solving synergistic optimization tasks simultaneously. However, the development of effective knowledge transfer mechanisms in MTO fundamentally relies on aligning elite solution distributions across tasks. This dependency creates a critical bottleneck in few-shot optimization regimes, as restricted evaluation budgets impede the identification of elite solution distributions required for beneficial transfer. This challenge is exacerbated in multiobjective multitask problems, where each optimizer must approximate a continuous Pareto manifold rather than a single optimal point. This paper introduces Iterative Sequential Transfer (IST) to circumvent this bottleneck. We model MTO as a sequence of sequential transfer optimization problems, concentrating evaluations on a single target per iteration. We propose a likelihood-informed task prioritization mechanism to maximize transfer utility by identifying the task most likely ready for knowledge integration. Empirical results on benchmark and real-world problems verify the effectiveness of the proposed method under tight budgets.

Optimising chemical reactions across multiple objectives, such as yield, selectivity, and safety, is central to chemical synthesis, and model-driven approaches depend critically on how reaction components are represented. Established featurisations are either chemically uninformative, as with one-hot encodings, or, as with molecular descriptors, do not readily extend across chemically distinct components. For structurally and functionally diverse components, it is therefore unclear what a shared representation should contain. Constructing such a representation is itself a challenging research undertaking that must be revisited for each new reaction system. Here we bypass this step by learning the reaction representation dynamically from text. Textual descriptions of reaction conditions are encoded by a fine-tuned language model trained jointly with Gaussian process surrogates, yielding task-adaptive representations within a multi-objective Bayesian optimisation loop. Across nickel- and palladium-catalysed cross-couplings in both sequential and parallel experimentation regimes, this approach reaches optimisation convergence in fewer experiments than descriptor libraries or one-hot encoding. Applied prospectively to a palladium-catalysed cyanation spanning mixed ligand denticity and heterogeneous additives, and to a three-objective asymmetric hydrogenation across chiral iridium and ruthenium catalyst families, two rounds of high-throughput experimentation (192 reactions, under 3% of each design space) delivered conditions translating directly to gram scale in 94% and 84% isolated yield, the latter at 99.6% enantiomeric excess.

The DIRECT algorithm is a deterministic global optimization method known for its versatility and balanced exploration-exploitation strategy. However, DIRECT-type algorithms are primarily effective for low-dimensional problems and often exhibit slow convergence as dimensionality increases, limiting their applicability to more complex optimization tasks. To address this limitation, this paper introduces X-DTC-GL, a novel DIRECT-type algorithm that incorporates dynamic partitioning and hybridization techniques. The dynamic partitioning approach adaptively refines the search space based on local one-dimensional surrogate models, enabling rapid subdivision of promising hyper-rectangles. The hybridization strategy selectively employs a hill-climbing method to exploit promising regions identified by the surrogate models. Extensive experiments on four diverse benchmark suites demonstrate that X-DTC-GL significantly outperforms existing DIRECT-type baselines, achieving improvements of ~12% in solvability and ~27% in solution quality. Performance-profile analyses indicate the fastest convergence on up to ~40% of instances, the best runtime performance on ~17% of problems, and competitive overall execution times. By improving performance within the partition-based framework, these advances strengthen the algorithm's competitiveness in state-of-the-art black-box optimization.

Real-world Multi-Objective Reinforcement Learning (MORL) often suffers from sparse rewards, reward conflicts, and late-stage reward tug-of-war, causing traditional linear scalarization to experience severe metric oscillations. To address optimization conflicts among multiple objectives in real-world deployment scenarios, we propose Multi-Marginal Preference Optimization (MMPO), a fine-grained framework that intervenes at the data, gradient, and constraint levels rather than relying on coarse-grained global scalarization. Specifically, MMPO performs exposure debiasing to mitigate sparse and biased rewards, applies priority-aware orthogonal projection to decouple conflicting gradients, and introduces self-prompted gradient constraints to prevent dominant objectives from overwhelming weaker ones. Experiments on real-world e-commerce datasets show that MMPO improves training stability and consistently achieves better performance across conflicting metrics. Moreover, it generalizes robustly to broader tasks such as ToolRL and code generation, demonstrating its effectiveness as a practical paradigm for multi-objective alignment.

Offline multi-objective optimization requires not only moving the objective vectors of candidate designs toward the Pareto front, but also distributing them effectively along it. Generative methods have recently emerged as a natural approach because they learn a distribution over feasible designs while allowing generation to be steered toward promising designs. Existing methods, however, largely retain classical sample-wise guidance strategies, leaving the distribution-level modeling capability of generative methods underused. We propose ParetoTransport, a training-free guidance method for pre-trained flow-matching models that explicitly specifies and refines a population-level distribution in objective space. ParetoTransport guides a flow-matching sampler to iteratively transport the empirical offline distribution toward the Pareto front, with Wasserstein matching to intermediate proxy distributions. This directly controls distributional displacement and mass allocation along the front. We establish a convergence result and demonstrate state-of-the-art performance on standard offline MOO benchmarks, extending recent evaluations beyond hypervolume to generational distance, inverted generational distance, and Wasserstein distance.

We introduce a novel Divide and Conquer control design methodology leveraging differential games in single-agent, multi-objective dynamical systems. The proposed framework associates each control objective with a virtual input and establishes a non-cooperative, finite or infinite horizon differential game among representative players. Each player optimizes a distinct virtual cost function tailored to its specific goal, the full system state, and the other virtual inputs, while accounting for the remaining players' optimal policies. By establishing a Nash Equilibrium for this game, we synthesize a composite controller that achieves a stable balance across competing objectives, providing control engineers with an intuitive and modular framework for parameter re-tuning throughout the design cycle. We provide formal mathematical derivations for both continuous-time and discrete-time dynamical systems, targeting large-scale single-agent applications where complex, dynamically conflicting control objectives make global weighting intractable. To demonstrate the methodology, we developed an open-source Python package implementing a novel numerical algorithm for solving Coupled Algebraic Riccati Equations arising in infinite-horizon differential games. We evaluate the approach on two benchmark case studies: an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor. The resulting closed-loop performance is compared against the classical Linear Quadratic Regulator (LQR) across various transient and steady-state control metrics, demonstrating superior trajectory tracking and robust multi-objective regulation.

Large language models are increasingly required to generate responses that satisfy multiple competing objectives. Since optimal trade-offs depend on both user preferences and input prompts, controllable multi-objective generation must dynamically adapt models at inference time without retraining. To address this, we propose Evolutionary Soups, a mixture-of-experts framework for fine-grained generation control, with gating networks trained via an evolutionary algorithm. The per-layer gating networks dynamically produce expert-merging coefficients from hidden-state representations, while the evolutionary algorithm incorporates greedy hypervolume contribution for effective evolution of these gating networks, achieving consistent improvements on large and noisy training datasets and broader coverage of the non-convex Pareto front. Experiments across three tasks demonstrate the effectiveness of Evolutionary Soups over baselines: it achieves the best hypervolume, linear utility, and Tchebyshev utility (~20% improvement) among controllable methods on all tasks.

Uncertainty quantification from finite data is central to machine learning, optimization, and automation systems, where decisions must remain reliable under limited samples and test-time distribution shift. Conformal prediction (CP) and distributionally robust optimization (DRO) offer two complementary approaches: CP constructs data-dependent prediction sets with distribution-free finite-sample validity under exchangeability, while DRO optimizes worst-case performance over an ambiguity set around an empirical distribution. We develop a unified probabilistic perspective on CP and DRO by viewing both as ways to turn finite calibration data into a data-dependent quantile estimator that a test score falls below with high probability. From this perspective, CP and DRO correct the empirical quantile along two coordinates of the same family of estimators: CP inflates the quantile level, whereas DRO shifts the quantile value through an ambiguity radius. Both methods provide the same calibration-conditional guarantee for the true distribution, requiring the target coverage to hold with high probability over the calibration sample. Their constructions differ, however: CP uses a closed-form, distribution-free level correction, while DRO uses a value-space correction whose certified radius depends on properties of the unknown distribution and additionally guarantees coverage uniformly over the ambiguity set. This distinction emerges in the tails of the score distribution. Because CP relies on sparse upper-tail order statistics of the calibration samples, its level inflation barely moves the estimator when those samples are dense near the target quantile but overshoots when they are sparse, whereas a well-chosen DRO radius corrects in value space and may avoid this overshoot.

Scientific discovery often involves optimising expensive-to-evaluate objectives over vast, structured, and open-ended hypothesis spaces, such as molecules, protein sequences, and computer programs. Generative models such as large language models (LLMs) provide expressive priors over such spaces, but their likelihoods and self-assessments are unreliable proxies for the objectives and calibrated epistemic uncertainty, especially for novel candidates outside the observed data distribution. We introduce the Large Discovery Model (LDM), an empirically grounded recurrent architecture that couples a generative model with a Bayesian non-parametric reward surrogate model. The generative model proposes and refines candidate designs, while the surrogate predicts their performance and quantifies uncertainty, yielding an uncertainty-aware value that guides candidate generation, refinement, and selection. The discovery memory and the surrogate model are continually updated as each new experimental observation arrives. We evaluate LDM on three scenarios spanning different design modalities and objectives, including neural-network training, antibody design, and molecular optimisation. Compared to LLM-only reflection or traditional statistical search across these domains, LDM achieves a $2.4\times$ greater reduction in validation BPB, an $18.2\%$ relative decrease in binding energy, and more than $60\%$ relative gains in molecular multi-objective performance. These results suggests that LDM could serve as a general-purpose discovery engine for effective search over open-ended hypothesis spaces.

Existing global optimization benchmark suites are of a moderate size and are based on a small number of analytical functions that date back even to the 1970s. This causes a risk of biasing the development of global optimization methods. We argue that the tasks related to the black-box adversarial attack (BBAA) can serve as valuable global optimization benchmark in many-dimensional space. We demonstrate the efficiency of several types of evolutionary algorithms and other metaheuristics in solving example BBAA problems. Thus, we take a step towards convergence of global optimization methods to the challenges and needs that arise in the modern machine learning field.

Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) to solve the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to convert the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method to solve the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.

We study nonnegative submodular maximization subject to a general matroid when the offline algorithm is given an arbitrary controlled value oracle. Our main result is an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson): without modifying its Poisson intensity, single-element exchange rule, or spiteful drop step, the algorithm retains limiting approximation factors $1/e$ for non-monotone objectives and $1-1/e$ for monotone objectives. More precisely, under every controlled oracle $\widehat f$ satisfying $|\widehat f(S)-f(S)|\le ξ$ for every set $S$, our implementation returns a feasible set with expected value at least $(1/e-\varepsilon)\OPT-O(kξ)$ and $(1-1/e-\varepsilon)\OPT-O(kξ)$, respectively, using $\widetilde O(nk^2\varepsilon^{-2})$ oracle calls. As a consequence, the offline-to-online reduction yields full-bandit CMAB algorithms for general matroid-constrained submodular rewards with exact limiting approximation-regret factors $1/e$ and $1-1/e$ and $\widetilde O(n^{1/5}k^{4/5}T^{4/5})$ regret.

Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral--nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of $\mathcal{O}(T^{-1/2})$ in the deterministic setting and $\mathcal{O}(T^{-1/4})$ under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Multi-objective Bayesian optimization (MOBO) is effective in identifying the Pareto fronts for expensive black-box problems. However, most current MOBO approaches are limited to low-dimensional decision space due to its exponential sampling complexity. This paper presents decision variable interaction analysis-based MOBO, ViaMOBO, a generic framework for expensive multi-objective problems with high-dimensional decision space. The key idea of ViaMOBO is that it utilizes a variable interaction analysis model to determine whether the decision space can be completely or partially divided, and then performs local Bayesian optimization in the divided decision subspaces. Through the variable analysis model, it can be derived whether the objectives in black-box problems are separable, partially separable, or non-separable based on the potential independent or interdependent relationships among decision variables without any strong assumptions. We compare ViaMOBO with the state-of-the-art MOBO methods on both synthetic and real-world benchmarks. The experimental results demonstrate that ViaMOBO outperforms other related MOBO baselines in approximating the Pareto front of high-dimensional expensive multi-objective problems.

Hit-to-lead optimization requires iterative design of hit analogs across competing potency, selectivity, physicochemical, pharmacokinetic, safety, and synthetic constraints. We present SABLE (Synthetically-accessible Agentic Bayesian Ligand Exploration), an open-source framework that employs natural-language orchestration to guide chemical structure optimization. SABLE uses an LLM to interpret user-defined goals and route tasks, while specialized tools perform reaction-templated analog enumeration, physicochemical and ADMET property prediction, structure-based affinity scoring, and Bayesian optimization. The resulting workflow is a computational twin of the analytical and prioritization stages of the design-make-test-analyze cycle, providing provenance of each numerical output. Across single, and multi-objective optimization studies, SABLE enriches candidate sets for user-defined computational objectives while evaluating only a subset of the enumerated search space. Its modular architecture allows tools and characterization backends to be replaced by editing a simple config file, without modifying operational logic. SABLE provides an extensible decision-support framework for prioritizing synthetically constrained analogs in early-stage drug discovery.

Evolutionary multitasking is a recent approach that solves multiple related optimization problems within a single evolutionary run, rather than addressing each problem separately. We consider monotone submodular optimization problems with dynamic knapsack constraints and study a multitasking formulation in which all tasks share a common monotone submodular function $f$, but differ in their constraints. We focus on the case where elements within each constraint have uniform cost and show that this structure leads to small Pareto fronts in the multitasking formulation. This enables solution sharing across tasks and can improve performance compared to running standard evolutionary approaches independently, depending on the constraint regime. Using rigorous runtime analysis, we analyze the expected time until the proposed multitasking algorithms obtain a $(1 - 1/e)$-approximation for each task. Experimental results for the Maximum Coverage problem complement the theoretical analysis and provide further insight into the practical behavior of the approach across different budget settings.

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.

Edge Artificial Intelligence (Edge AI) enables the deployment of AI models directly on local edge devices, while such deployments are subject to strict resource constraints, particularly in clinical applications requiring local and timely inference. In such contexts, explainable artificial intelligence (XAI) can serve as a human-AI interface intended to support healthcare professionals' and patients' understanding of model predictions and informed decision-making. To fulfill this role, XAI method selection for TinyML deployments can be formulated as a human-centered multi-objective design problem that jointly considers qualitative stakeholder preferences, explanation quality, and proxy-based deployment cost. We propose a framework that integrates a large language model (LLM)-guided design interface that maps qualitative stakeholder preferences to candidate XAI methods, followed by deterministic feasibility filtering and Pareto-based optimization. The framework exposes trade-offs among explanation fidelity, stability, and proxy-based deployment cost while characterizing their implications for explanation quality and estimated deployment feasibility. A proof-of-concept evaluation on a skin lesion classification task illustrates how the framework systematically compares candidate XAI methods and identifies Pareto-efficient trade-offs. The present evaluation covers the computational selection stages, while physical MCU deployment and empirical human-expert validation remain outside the scope of this study.

To address the issue that existing dynamic multi-objective optimization algorithms mainly rely on individual migration or independent special point sampling after environmental changes, while failing to fully exploit the structural relationships among representative solutions, a Special Point Skeleton Reconstruction based Dynamic Multi-Objective Evolutionary Algorithm (SPSR-DMOEA) is proposed. First, the centroid, knee points, and extreme points are extracted from the Pareto optimal solution set of the current environment, and their positions in the new environment are adaptively predicted according to their movement velocities across consecutive environments. Subsequently, in the decision space, the centroid is connected with other anchor points, and a minimum spanning tree is constructed among the non-centroid anchor points, thereby establishing a prediction skeleton capable of describing the overall population structure. According to the lengths of the skeleton edges, the number of individuals allocated to each edge is determined proportionally. Candidate solutions are uniformly generated along each edge, and random orthogonal perturbations are introduced to expand the search region around the skeleton. Experimental results on the DF dynamic multi-objective benchmark suite demonstrate the effectiveness of the proposed method in dynamic tracking capability.

Analog circuit design automation using reinforcement learning (RL) has emerged as a promising approach for reducing manual effort. However, many existing RL-based methods focus on single-objective optimization. Even methods designed for multi-objective (MO) problems often reduce multiple design specifications to a single scalar reward. This simplification limits the ability to capture the true Pareto trade-off among competing objectives and often leads to suboptimal designs. Moreover, requiring the model to be retrained from scratch whenever the desired MO specifications change remains a key limitation. To address these challenges, we present ORACLE, an open-source RL-based framework for MO analog circuit design optimization that replaces scalar reward optimization with vector-valued learning and preference-aware conditioning. ORACLE represents a true MO analog circuit design optimizer that uses a preference vector to specify the relative weights of multiple objectives, enabling a single trained model to generate designs across diverse trade-off settings without retraining. We further propose two preference-guidance strategies, namely normalized-weight guidance and cosine-aligned guidance, to improve convergence. In addition, we incorporate a large language model (LLM)-guided action selection mechanism to filter actions that are likely to lead to suboptimal designs or increased runtime. Our results show that, on multiple circuit topologies with 2,000 test cases, ORACLE reduces runtime by 20.4x - 104.4x compared to state-of-the-art approaches. It also meets 99.9% of the 2,000 target specifications, and achieves 5.1x - 318.6x better figure of merit in the resulting output specs.

Advanced materials discovery increasingly relies on machine learning and Bayesian optimization to explore large discrete design spaces under limited evaluation budgets. However, conventional Bayesian optimization (BO) can become inefficient as candidate spaces grow, often evaluating low-value regions before reaching informative areas. We propose an active-learning (AL)-guided adaptive search-space refinement framework combined with multi-objective BO to accelerate materials optimization while preserving Pareto-relevant regions. We evaluate the approach on CH4/N2 separation in covalent-organic frameworks and pressure-vessel design with material-direction stress components and thickness objectives. Results show that the AL-guided refinement reduces the candidate space by approximately half while preserving more than 99 percent of the original hypervolume. The reduced-space strategy improves early convergence and cumulative Pareto-front discovery from the BO, demonstrating efficient large-scale materials optimization across constrained autonomous materials discovery settings.

Robotic planning often involves multiple objectives with complex priority relationships, such as safety, efficiency, and regulatory compliance. Rulebooks formalize these relationships, allowing partial ordering of objectives that generalizes both Pareto and lexicographic dominance. Computing the full set of rulebook-optimal solutions, however, is computationally expensive. To address this challenge, we introduce the concept of epsilon-rule-dominance, a principled notion of approximate dominance under rulebooks, and propose RApex, a best-first search algorithm that efficiently computes a compact set of epsilon-approximate rulebook-optimal solutions. RApex leverages dimensionality reduction, a technique used to speed up existing multi-objective search algorithms, while respecting rule hierarchies by maintaining separate closed sets and performing dominance checks over truncated and residual rule sets. We provide a formal analysis of RA*pex, proving that every rulebook-optimal solution is epsilon-rule-dominated (a generalization of approximate dominance we introduce) by at least one solution in the returned set. Empirical results demonstrate that our approach achieves computation times over two orders of magnitude faster than existing methods.

Black-box optimization is a ubiquitous problem in science and engineering, often dealing with expensive objective functions with cheaper lower-fidelity proxies available. Multi-fidelity Bayesian optimization (MF-BO) is a principled approach to this problem, leveraging correlations across different fidelities when querying the objective. However, for many important MF-BO tasks, the true highest-fidelity function is prohibitively expensive to be part of the optimization loop. Nevertheless, practitioners often have gold standard data (observations of the highest-fidelity function) obtained from previous experiments that might provide information for the current task. For instance, in molecular optimization, chemists often pick the top-$k$ candidate molecules using various computer simulations, and later reveal their true objective function values. In this work, we demonstrate the suboptimality of standard MF-BO algorithms in the real-world scenarios above, even under ideal assumptions. Next, we mitigate this problem by incorporating historical high-fidelity data accompanied by task descriptors---which can be explicitly given or extracted from unstructured metadata. We demonstrate the effectiveness of our methods on synthetic functions, as well as real-world problems in chemistry and hyperparameter optimization.

This work revealed a relationship between a multi-objective optimization problem and single-objective optimization problems that exist in the multi-objective problem. This work focused on combinatorial problems and investigated the relations between the local optima networks of the single-objective problems and the Pareto optima network of the multi-objective problem. Each of their networks has a graph structure. We divided the entire network into subgraphs. Each subgraph was called a component and characterized by overlapping relations between the single-objective local optima networks and the multi-objective Pareto optima network. Results on multi-objective landscape problems showed that most Pareto optimal solutions were reachable from the single-objective local optimal solutions. This tendency was emphasized by increasing the number of objectives and the objective correlation. The number of co-variables impacted the number of cross-link relations between the single-objective local optima networks and the multi-objective Pareto optima network. The results suggested that searching for single-objective problems is a clue to multi-objective optimization.

We consider grey-box optimization problems where the decision variables naturally partition into black-box variables (as arguments to an expensive black-box function) and white-box variables, governed by a set of explicit, closed-form equations that also depend on the output of the black-box function. We exploit this separability through a bilevel reformulation: an outer Bayesian optimization (BO) to optimize the scalar objective as a function of black-box variables alone, while an inner problem solves the white-box subproblem via global optimization. The Gaussian process surrogate used in BO is therefore defined rather than and white-box constraints are satisfied exactly whenever the inner optimizer converges to a feasible point---without penalty functions, chance constraints, or moment approximations. On a suite of 13 benchmark problems, bilevel BO achieves lower regret, with fewer iterations and wall clock time. This advantage is robust to initialization set size, exploration parameters, and inner-solver choice.

We investigate Optimization under Input Uncertainty (OIU), in which the input to the objective function, rather than the objective function itself, is subject to uncertainty. OIU appears in manufacturing processes with production tolerance, control of physical systems with actuation noise, Mixture of Experts, and Reinforcement Learning (RL). Most of the existing approaches solve OIU by using the value of the objective function but discard the information of the realized input, even though the realized input is observable in various applications. The question here is whether the discarded information of the realized input is useful to accelerate the optimization process. We affirmatively answer this question for Evolutionary Strategy (ES) by theoretically showing that the information of the realized input can reduce the variance of the gradient estimator via Rao-Blackwellization. Using the Rao-Blackwellized gradient estimator, we propose Phenotype-Accelerated Evolutionary Strategy (PAES), which is a refinement of ES for OIU. Numerical experiments show that PAES converges faster than the usual ES from simple continuous optimization problems to RL benchmarks.

Polymer property prediction and inverse generative design targeting desired properties are two crucial tasks in machine learning-assisted polymer design. While the former has received considerable attention, there have been limited methods developed for the latter. Existing methods focus on single-property optimization in the generative process, whereas accurate prediction of macroscopic material behavior requires simultaneous control of multiple physical properties. In this paper, we provide a transformative framework for direct optimization of a large collection of polymer properties. We propose PolymerGPT, a decoder-based GPT model that incorporates up to 37 commonly used polymer properties into the generative process via learned conditioning prefixes. It also supports a scaffold condition that specifies a desired scaffold for predicted structures. Our experimental results demonstrate that PolymerGPT achieves exceptional performance for unconditional and conditional generation while maintaining high validity, uniqueness, and novelty. Conditioning on five key properties yields generated structures whose predicted values closely match all target properties simultaneously.

Streaming data-driven dynamic multi-objective optimization requires algorithms to track time-varying Pareto fronts using only sequential observations under concept drift. However, systematic evaluation remains difficult because real-world problems usually lack ground-truth optima, drift annotations, and controllable conditions, while existing benchmarks provide limited support for standardized comparison. This paper proposes SDDMO-Bench, a benchmark suite that transforms classical dynamic multi-objective test problems into streaming environments by combining intrinsic objective-mapping evolution, controllable distributional drift, and sequential data revelation. By combining five representative time-dependent base functions with six distributional drift patterns, SDDMO-Bench constructs 30 scenarios with diverse levels of non-stationarity, problem complexity, sample-distribution variation, and Pareto-front evolution. Experiments with representative evolutionary algorithms demonstrate that SDDMO-Bench provides challenging and discriminative test scenarios, offering a standardized, controllable, and reproducible benchmark for evaluating adaptability, robustness, and Pareto-front tracking in streaming data-driven dynamic multi-objective optimization.

We derive finite-sample generalization bounds for Transformers trained with dynamic programming recursions. Building on the doubly lifted, measure-valued formulation of Transformer dynamics, we view data sets as probability laws on pairs of empirical input-output measures, allowing us to interpret the training problem as a finite-horizon Markovian control problem. We then analyze a quantized model, derived by quantizing the state, action, and measure-state spaces, and derive explicit finite-sample generalization bounds using concentration inequalities for empirical laws on finite metric spaces together with a Lipschitz stability estimate for the value function. These bounds are transferred to the base model at the cost of an explicit approximation error. Finally, we show that the same machinery yields a distributionally robust control formulation of the training problem, connecting Transformer generalization to Wasserstein distributionally robust optimization.

Robust Optimization Over Time (ROOT) is a recent branch of evolutionary dynamic optimization that seeks solutions capable of remaining effective across multiple consecutive environments. Unlike the traditional track-the-moving-optimum (TMO) paradigm, which reoptimizes after every environmental change, ROOT explicitly values persistence. Although the field has grown considerably, most contributions remain algorithmic and empirical, leaving several fundamental properties poorly understood from a theoretical perspective. One such property is survival time, defined as the number of future environments in which a deployed solution continues to satisfy a prescribed quality threshold. While survival time is widely used as a measure of temporal robustness, little is known about how its expected value depends on environmental dynamics, deployment quality, or problem characteristics. This paper studies expected survival time for a fixed deployed solution under isotropic Gaussian environmental dynamics. Modeling survival as a discrete first-exit problem, we derive a rigorous lower bound and a computable multi-step upper bound. The analysis shows that expected survival scales as $Θ(σ^-{2})$ in slowly varying environments and approaches its minimum value of one future change in high dimensions. A comprehensive Monte Carlo study validates the theoretical predictions, examines sensitivity to modeling assumptions and parameter uncertainty, and illustrates how the bounds can support deployment decisions after optimization. The resulting framework provides an analytical characterization of deployment lifetime and identifies when a required deployment horizon can be guaranteed, ruled out, or remains analytically unresolved.

Generative models are increasingly adopted in distributionally robust optimization (DRO), but existing approaches trade off model compatibility and adversarial structure: methods that accept arbitrary samplers do not restrict worst-case laws to a generator family, while generator-parameterized adversaries rely on model-specific access such as likelihoods, scores, or training data. We propose Generative Distributionally Robust Optimization (GDRO), a principled framework that accepts any sampleable conditional generator as the nominal model and restricts worst-case laws to a chosen conditional generator family. The key is the sampler-Sinkhorn pairing: samplers represent the conditional laws exactly, while Sinkhorn divergence compares their induced distributions without likelihood access and can be estimated from samples alone. The resulting population problem admits a direct finite-sample approximation and differentiable primal-dual implementation at the active decision context. For Lipschitz losses, the population Sinkhorn radius bounds downstream degradation. Across explicit and implicit generators, our method reduces rare-context inventory regret by 60% and SocialGAN navigation collisions by 50% relative to nominal decisions.

This paper investigates the role of dynamic population sizes in evolutionary multi-objective optimization. Although such approaches are widely used in practice, their benefits remain poorly understood, and rigorous runtime analyses explaining when and why they help are still scarce. To address this, we introduce the bi-objective problem class CLIMB and analyze the runtime of GSEMO and the widely used NSGA-II on this problem. Our results show that allowing a dynamic population size for NSGA-II can lead to a moderate improvement, yielding a speedup of order $Ω(\sqrt{n}/\log n)$. In particular, we prove that GSEMO and NSGA-II-DYN, a version of NSGA-II with dynamic population sizes we propose in this paper, can find the Pareto front of CLIMB in expected $O(n \log n)$ fitness evaluations, whereas NSGA-II with a fixed population size requires $Ω(n^{1.5})$ fitness evaluations in expectation. To the best of our knowledge, this is the first rigorous runtime analysis in multi-objective optimization demonstrating a super-constant speedup of GSEMO over NSGA-II. Our analysis builds on concepts from single-objective optimization, like the evolution of population diversity over time, and employs the well-known family-three method to prove the lower bound.

In this paper, we propose and study a robust variant of the smart predict-then-optimize approach that accounts for prediction shifts due to disturbance in the covariate feature space. While traditional integrated-learning-and-optimization models assume that side information is perfectly revealed, empirical data-driven features are frequently corrupted or noisy at the time of decision-making, leading to fragile operational policies. To bridge this gap, we integrate principles of robust optimization directly into the predictive-prescriptive pipeline via a smart predict-then-robustly optimize loss and establish a computationally tractable convex surrogate, designed to hedge against worst-case feature perturbations. On the theoretical front, we formalize the structural validity of this surrogate by proving its approximation error probability decays exponentially according to a sub-Gaussian concentration profile. Furthermore, we establish that under mild assumptions, the surrogate is Fisher consistent with high probability. We also prove necessary conditions under which our framework outperforms standard smart predict-then-optimize and maintain its superiority even when the standard method is equipped with regularized upstream predictions. Numerical experiments validate that our robust framework consistently yields significant performance improvements over standard methods, both in out-of-sample terms and in training stability.

Cooperative multi-task semantic communication (CMT-SemCom) improves task execution performance by leveraging shared representations. However, as we demonstrated in [1], cooperative multi-tasking can be either constructive or destructive, depending on the semantic relationships among tasks. To ensure constructive cooperation, we propose a semantic-aware task clustering method for CMT-SemCom. We have formulated a sequential multi-stage optimization problem in which semantically aligned tasks are clustered once after a short initial training phase, and then end-to-end (E2E) joint training is conducted exclusively within the discovered groups. Specifically, the problem decomposes into two stages: (i) a semantic clustering problem leveraging hierarchical density-based spatial clustering, and (ii) an intra-cluster E2E CMT-SemCom learning problem. Simulation results demonstrate that the proposed framework effectively mitigates destructive cooperation and negative transfer, yielding accuracy gains compared to unclustered multi-tasking and individual training baselines.

We study distributionally robust linear chance-constrained problems in which uncertainty is modeled by a Gaussian mixture model (GMM). Finite-support distributionally robust (FDR) formulations, widely used in data-driven robust optimization, robustify over empirical mixture support points and therefore primarily stress-test the fitted nominal mixture. This can be insufficient when service reliability depends on structural misspecification of the nominal mixture-support parameters. To address this limitation, we describe the ambiguity set of distributions by developing a novel formulation of a Wasserstein-2 metric that uses the Bures-Wasserstein (BW) metric over probability measures with finite second moments. Unlike FDR, which generally sets finitely many empirical support points a priori, the proposed ambiguity set allows the worst-case distribution to endogenously determine both how many mixture components receive mass and where their means and covariances lie within a continuous support. For the resulting ambiguity set, under mild regularity conditions, we prove strong duality for the inner worst-case chance-constraint problem and derive its semi-infinite reformulation. We then develop an adaptive cutting-surface algorithm, which endogenously determines the locations of mixture components receiving mass, and the mean and covariances of the Gaussian distributions at these locations. The algorithm attains any prescribed optimality gap in finitely many iterations, while a block-alternating local search identifies new components. A case study using the electric-vehicle charging-station energy-allocation problem demonstrates the framework's practical value in achieving any reliability targets. CDR also induces structural changes in energy allocations, unlike FDR, whose allocations remain close to the nominal solution.

Large-scale multi-objective optimization problems (LSMOPs) are challenging due to their high-dimensional decision spaces. Fuzzy search is an effective technique for improving search efficiency, while decision variable analysis can reveal the distinct roles of variables in promoting convergence and maintaining diversity. However, existing fuzzy search methods generally employ a uniform search granularity for all variables, overlooking the heterogeneous search requirements implied by variable roles. To address this limitation, this paper proposes a Decision variable analysis-guided Differentiated Fuzzy Search method, termed DDFS. The proposed method establishes an explicit mapping between decision-variable roles and fuzzy search granularities. Decision variable analysis is employed to identify variable roles and search sensitivities, enabling different variable groups to adopt differentiated fuzzy search behaviors during offspring generation. Furthermore, a Dual-Indicator Stage Transition Mechanism is developed to dynamically adjust fuzzy-updating intensity throughout the evolutionary process, balancing early-stage search-space compression and late-stage convergence refinement. Extensive experiments on the LSMOP and UF benchmark suites with up to 1000 decision variables show that DDFS generally achieves competitive performance against several representative large-scale multi-objective evolutionary algorithms. The results suggest that explicitly incorporating decision-variable roles into fuzzy search can help improve optimization performance in high-dimensional decision spaces.

Automated optimisation is increasingly adopted in industrial processes, yet a trust gap persists between engineers who design these algorithms and operators who must act on their recommendations. Explainable AI methods like SHAP (SHapley Additive exPlanations) have transformed interpretability for machine learning predictions; optimisation outputs could benefit from similar techniques. We present an approach that integrates Implicit Function Theorem (IFT) based sensitivity analysis with SHAP attribution and narrative generation via Large Language Models (LLM), producing explanations tailored for operators. Our approach leverages IFT to compute exact parameter sensitivities $\partial p^*/\partial x$ from the optimality conditions, enabling efficient GradientSHAP computation. For an industrial High Pressure Grinding Roll (HPGR) control optimisation problem with 22 features, we achieve equivalent SHAP attributions (correlation $>$0.99 with KernelSHAP) with over 40$\times$ speedup, enabling real-time natural language explanations. We validate on industrial scenarios and present feedback from domain experts on generated explanations.

Many signal processing systems ultimately exist to {act}. Whenever the state variable that determines the action to be taken by a decision maker, or agent, is uncertain, the way that uncertainty is represented decides how well the agent performs and how much its performance can be trusted. This lecture note develops, from first principles and within a single decision-theoretic setting, the link between the {objective} and the knowledge of an agent and the form of uncertainty representation that is sufficient to act optimally. To start, assuming a known environment distribution, we show that a risk-neutral agent needs the posterior distribution over the state, whereas a risk-averse agent can rely without loss of optimality on a {prediction set} and a worst-case decision rule. We then turn to the case in which the environment is unknown, and identify three complementary approaches to address the resulting epistemic uncertainty: calibration of a fixed predictor, credal (ambiguity) sets with distributionally robust optimization, and Bayesian inference over model parameters. The common thread is that reliable decisions require an uncertainty representation matched to the decision objective and to the knowledge profile of the agent, together with a guarantee that certifies the utility the agent will actually obtain.

While Large Language Models (LLMs) excel in many general NLP tasks, their formal reasoning capabilities are often compromised by content effects, demonstrating a measurable bias towards real-world plausibility. In this paper, we present our system for SemEval-2026 Task 11, which evaluates the ability of models to disentangle formal logic from content across 12 languages with and without distractor premises. We address this challenge using mDeBERTa-v3 networks fine-tuned on a synthetic, rule-based dataset of syllogistic schemes to avoid the semantic noise of LLM-augmented data. To explicitly decouple plausibility from logical structure, our training pipeline employs a multi-objective loss function combining Adaptive Group Distributionally Robust Optimization (DRO), a scheduled differentiable bias penalty, and KL-Divergence consistency regularization. Our system achieved #1 ranks and perfect Ranking Scores (100.0) with 0.00% bias and 100.0% accuracy on Subtask 1 (English), Subtask 2 (Noisy English), and Subtask 3 (Multilingual). On the highly complex Subtask 4 (Noisy Multilingual), the system achieved the 6th rank with 89.06% Accuracy and F1-score, alongside a limited 2.89% Bias and a 37.78 Ranking Score. Our dataset generation engine and codebase are publicly available to facilitate future work on robust logical reasoning.

This paper introduces the Dynamical Vehicle Orienteering Problem (DVOP), a generalization of the Orienteering Problem (OP). The OP maximizes the reward collected from spatial targets under a limited travel budget; the DVOP extends it by accounting for both external and vehicle-actuated forces. We study the DVOP in the context of multi-rotor Unmanned Aerial Vehicle (UAV) flight planning, using a three-dimensional Point-Mass Model (PMM) constrained by maximum velocity and acceleration magnitudes and subject to gravitational acceleration, with the travel budget expressed as a maximum flight time. Because the DVOP couples reward maximization with time-optimal trajectory planning, it cannot be formulated as a simple graph problem and solved exactly without relaxing or under-actuating the vehicle dynamics. We therefore propose two solution approaches: a Branch-and-Bound (BnB) procedure that combines Non-Linear Programming (NLP) and Mixed-Integer Linear Programming (MILP) to provide high-quality solutions, and a Large Neighborhood Search (LNS) metaheuristic that supplies an initial reward bound and scales to instances intractable for the BnB. The BnB relies on a novel MILP formulation of travel costs based on minimum-time trajectory primitives through target triplets, yielding a tight reward upper bound, while the LNS uses limited thrust decomposition to compute fast, high-quality PMM trajectories. Experiments on benchmark instances show improvements of up to 37 % over state-of-the-art solutions for the Kinematic Orienteering Problem, and a real-world deployment on a multi-rotor UAV verifies the proposed PMM solution trajectories.

The rapid deployment of machine learning systems across cloud, edge, and enterprise environments has brought model optimization to the forefront of systems-engineering. Despite a rich literature spanning quantization, pruning, knowledge distillation, parameter-efficient fine-tuning (PEFT), and inference-time optimization, practitioners are often left navigating these techniques through heuristics rather than principled methodology. We argue that optimization should be formulated as a constraint-driven, multi-objective engineering decision and introduce a unified framework that characterizes any production deployment along five interacting constraint dimensions: data availability, latency budget, memory budget, accuracy tolerance, and retraining budget. Building on this taxonomy, we synthesize empirical gains reported across the research literature and map them to operational constraints rather than algorithmic categories. To ensure practical relevance, we selected these techniques by reviewing recent literature for methods that report measurable improvements against critical deployment bottlenecks. We propose a prescriptive decision framework and provide optimization pipelines for four representative industrial scenarios to illustrate it in practice. To the best of our knowledge, this work provides one of the first structured attempts to formalize model optimization as a constraint-aware, multi-objective engineering process, synthesizing quantitative evidence from the research literature.

Many design tasks can be cast as black-box function optimization, enabling use of Bayesian optimization to find an ideal design with minimal number of trials. However, often we do not actually need the optimum but instead a sufficiently good solution is enough, for instance a material that is durable enough for its intended use. In most cases there are multiple satisfactory solutions, forming a superlevel set of the function, raising a key question of which one to prefer. We answer this by explaining why robustness to input perturbations that may occur when the solution is deployed is a good criterion and by introduce a Bayesian optimization method that efficiently finds satisficing solutions that are robust to maximally large perturbations. In contrast to previous works, we assume the inputs can be accurately controlled during optimization, but will be perturbed after the deployment.

Many-objective optimization has been a field of interest over the past two decades and several evolutionary optimization algorithms have been introduced to tackle these problems; yet two fundamental questions remain underexplored: (i) What happens when the number of objectives grows beyond the typical many-objective regime of about fifteen and becomes massive? (ii) How do problem characteristics, such as the nature of interactions between objectives, influence algorithmic performance? To answer these questions we employ a diagnostic benchmark suite that allows control over problem characteristics and can be scaled to extremely high objective counts. Using this framework we evaluate several state-of-the-art evolutionary algorithms including NSGA-II, NSGA-III, MOEA/D and lexicase selection across a range of dimensionalities and diagnostic problem landscapes. Our experiments reveal that problem characteristics significantly affect algorithm performance. In particular, the nature of interactions between objectives appears important. These results highlight the importance of understanding these properties before selecting an algorithm for a specific problem. We also show that lexicase selection, an algorithm originally designed for genetic programming, compares favorably with state-of-the-art many-objective optimization algorithms while avoiding the dependence on predefined reference directions.

Predict-then-optimize systems usually compress uncertainty into a point forecast and then solve a downstream optimization problem as if the forecast were reliable. Distributionally robust optimization (DRO) offers protection against misspecification, but the ambiguity set is often centered at historical samples and uses a fixed radius. We propose \emph{learned predictive ambiguity sets} (LPAS): a deep contextual model outputs a finite nominal scenario distribution, a state-dependent Wasserstein radius, and optionally an anisotropic ground metric. These outputs define a contextual ambiguity set that feeds a DRO decision layer. The radius is trained by a combination of conditional quantile calibration, size regularization, and downstream decision loss, so that robustness is adaptive rather than globally fixed. We derive the finite dual form used by the decision layer, present a staged training algorithm, and evaluate the method on distributionally robust portfolio optimization with 20 S&P 500 constituents from 2018--2026. The proposed method substantially improves over equal-weight, predict-then-optimize, and historical Wasserstein DRO baselines, achieving 26.28% annualized return, Sharpe ratio 1.30, final wealth 1.61, and lower tail loss than a deep fixed-radius DRO baseline while using a smaller average radius. The results show that learned ambiguity radii can recover most of the performance of strong fixed-radius DRO while reducing unnecessary conservatism and improving regime adaptivity.

Large-scale sparse multiobjective optimization problems (LSSMOPs) involve a large number of decision variables and Pareto optimal solutions with only a few nonzero variables. However, as the number of decision variables grows, it becomes increasingly challenging to accurately identify the nonzero variables, and optimization performance is adversely affected. To address these issues, this paper proposes an evolutionary algorithm for LSSMOPs. Specifically, we propose a new initialization method capable of generating scores that accurately reflect the importance of variables, and an initial mask vector template that can locate nonzero variables. This leads to the generation of a high-quality initial population. Additionally, this paper introduces a new strategy to calculate the mutation probability for each variable and a novel optimization for real variables based on the Pareto-guided normal distribution, enabling the population to avoid being trapped in local optima and quickly converge to the global optimum. Experimental results from eight benchmark problems and three real-world applications demonstrate that the proposed algorithm achieves superior performance compared with state-of-the-art algorithms.

Multi-objective masked image modeling (MIM) combines complementary learning signals (token distillation, CLS alignment, and pixel reconstruction) but existing methods weight these objectives with global scalars, ignoring spatial heterogeneity across patches. We present ExPLoRe (Expert Patch-Level Loss Routing), which repurposes Soft Mixture of Experts (MoE) dispatch weights as learned, per-patch loss coefficients. The key mechanism is loss-coupling: allowing loss gradients to flow through dispatch weights to the router enables content-dependent specialization, where different patches receive different emphases across objectives. A detach ablation confirms loss-coupling as the core mechanism, degrading performance by 1.6% when gradients are blocked. On ImageNet-1K with ViT-Base, ExPLoRe improves over non-MoE baselines on two objective combinations (Token+CLS: +0.5% k-NN, +4.4% linear probe; Token+Pixel: +2.2% k-NN), achieving 80.6% linear probe and 85.3% finetuning accuracy, competitive with published methods. For downstream transfer, we develop adaptation recipes (Freeze Routing, Expert Dropout, and Freeze Attention) that improve MoE finetuning by +1.5% over the vanilla MoE, and close a 2.5--2.9 mIoU segmentation gap so that MoE models match or exceed non-MoE baselines on ADE20K.

Optimizing nonlinear preferences in multi-objective reinforcement learning (MORL) is essential for capturing complex trade-offs like risk aversion or fairness. However, such non-linearity has historically bifurcated nonlinear MORL objectives into two distinct paradigms: Scalarized Expected Return (SER) and Expected Scalarized Return (ESR). While SER requires global-level optimization and ESR requires non-Markovian policies, leading to fragmented optimization strategies, we bridge this divide through the Aggregation-Expectation-Transformation (AET) framework. By unifying both criteria through a tripartite decomposition of scalarization, AET provides a principled foundation for general nonlinear MORL. Building on this framework, we propose AETDICE, a tractable offline RL algorithm for AET objectives. By utilizing DICE-style density-ratio estimation in an augmented state space, AETDICE enables sample-based optimization from static datasets. Our framework resolves long-standing barriers and captures respective trade-offs induced by AET framework, which existing methods fail to address.

Explainability in Many-Objective Optimization (MaO) is currently hindered by the escalating complexity of the Pareto front, which renders the relationship between high-dimensional decision variables and objective outcomes increasingly opaque. As the number of objectives exceeds the limits of traditional visualization, decision-makers encounter a cognitive drought'' in identifying relevant trade-offs or specifying target regions without a priori knowledge. To bridge this interpretability gap, we introduce the {Partition-Guided Distance Saliency (PGDS)} framework, a novel XAI approach designed for continuous optimization landscapes. Our framework automates the explanation process through a three-stage pipeline that prioritizes geometric intuition over abstract rules. First, we employ a surrogate model that learns how geometric distances in the decision space map to proximity in the objective space. Second, to address the difficulty of manual target selection in high dimensions, the framework automatically partitions the objective landscape into distinct regions and identifies local Dominating Points'' to serve as automated targets for improvement. Third, we quantify how sensitive a solution's position is to each decision variable by measuring the distance shifts induced by perturbations to each variable. This allows PGDS to categorize features as either Drivers'' which facilitate convergence toward preferred regions, or Blockers'' which represent geometric constraints hindering further progress. Validation on 10-objective benchmarks and a physics-informed engineering problem (Welded Beam) demonstrates that PGDS provides differentiated, actionable insights that traditional visualization and rule-based XAI methods fail to provide.

The continuous integral R2 indicator is a Pareto-compliant refinement of the classical finite-weight-vector R2 indicator, used in performance assessment, bounded archiving for a-posteriori multi-objective optimization, and skyline selection in databases. This work introduces a bidirectional perspective mapping between continuous integral R2 computation and integration over unions of anchored axis-aligned boxes. After translating the ideal point of a minimization problem to the origin, approximation points become strictly positive loss vectors, and the subgraph of the lower weighted Tchebycheff envelope over the weight simplex maps to the complement of an anchored-box union in reciprocal objective space. The Jacobian gives an absolute R2 formula as a weighted complement volume with density $(x_1+\cdots+x_N)^{-(N+1)}$, while differences of R2 values become finite weighted hypervolume differences. Hence, hypervolume algorithms that emit box decompositions can be reused by replacing ordinary box volumes with closed-form weighted box integrals. For $N$ objectives, this gives an output-sensitive overhead $O(2^N M)$ for an $M$-box decomposition, or $O(M)$ for fixed $N$. Using existing box-decomposition approaches, the integral R2 can be computed in $O(n \log n)$ for $N=2,3$, in $O(n^2)$ for $N=4$, and in $O\left(n^{\lfloor (N-1)/2\rfloor+1}\right)$ for $N\geq4$, with $n$ denoting the size of the approximation set. On the lower-bound side, exact value computation has an $Ω(n\log n)$ lower bound in the algebraic decision-tree model already in two objectives, this bound lifts to every fixed $N\geq2$, and exact computation is $\#P$-hard when $N$ is part of the input. Together, the proposed perspective mapping provides a powerful tool for transferring algorithmic and structural results between anchored-box union and hypervolume theory and integral R2 computation.

Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as $P(Y|X)$, our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.

Humans facing algorithmic decision systems have been found to ``game'' them by altering their input data (at a cost to them) in order to favorably change the algorithmic outcomes they receive (at a cost to the algorithm). The growing literature on strategic classification seeks to develop robust machine learning algorithms that account for, and reduce, unwanted strategic behavior. A limitation of these existing works is that they assume the cost of strategic behavior to be fixed and independent of the classifier's decision. In practice, however, manipulation costs evolve and depend on past algorithmic decisions: today's decisions influence tomorrow's costs. This paper proposes and analyzes a two-stage robust optimization framework with a decision-dependent uncertainty set to capture such dependencies. We highlight that awareness of policy-dependent costs not only reduces uncertainty, but also better curtails gaming of the algorithmic system over time.

Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $τ\in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $τ$ provides.

Scalar objective functions are required when a multi-criteria optimization problem must yield a single preferred design rather than only a Pareto set. The choice of scalarization influences which compromise is selected, how preference parameters are interpreted, and whether non-supported Pareto regions can be reached. This paper compares four formulations for normalized bi-criteria minimization: weighted sums, achievement scalarizing functions, desirability functions, and a fuzzy-logic-based formulation. Two analytically defined Pareto fronts, one convex and one concave, isolate the effect of the objective formulation from numerical optimizer behavior. The comparison focuses on reachable Pareto regions, parameter-induced selection density, compensation between criteria, sensitivity, and interpretability. Results show that weighted sums are simple but structurally limited on concave fronts, while achievement, desirability, and fuzzy formulations reach interior non-supported regions through different mechanisms. Desirability functions introduce nonlinear single-criterion preference mappings, whereas fuzzy rules express nonseparable and reference-dependent engineering preferences.

Selecting a fixed number of representative points from a finite Pareto-front approximation is a fundamental post-processing task in multiobjective optimization. This paper studies this problem for the integral R2 indicator in three objectives, where the indicator is defined as the integral of the lower envelope of weighted Tchebycheff scalarizations over the two-dimensional weight simplex. We provide two complementary algorithmic results. On the positive side, we show that the integral R2 improvement with respect to any fixed baseline is a monotone submodular set function. For the usual ideal-point based R2 indicator, with the ideal point fixed, this yields a direct gap-reduction guarantee: greedy selection closes at least a $(1-1/e)$-fraction of the maximum possible R2 gap between a fixed dominated anchor value and the best cardinality-$k$ value. We also give a tested greedy implementation that evaluates exact integral R2 values by subdivision, with worst-case running time $O(n^6)$. On the negative side, we prove that exact fixed-cardinality subset selection is NP-hard already in three objectives. The hardness proof uses a perspective transformation that maps Tchebycheff-shadow improvements to a weighted anchored-box union problem with density $(x_1+x_2+x_3)^{-4}$, and then adapts the three-dimensional anchored-box construction of Bringmann, Cabello, and Emmerich. Together, these results separate the tractable two-objective case from the three-objective case while identifying a principled approximation route based on submodular optimization.

Classical uncertainty sets for robust optimisation impose fixed geometric shapes that cannot represent the complex dependencies present in real-world data. We propose Generative Robust Optimisation (GRO), a framework in which a deep generative model defines the uncertainty set as the image of a neural network decoder over a calibrated latent set, naturally accommodating nonlinear correlations, asymmetry, and multimodality. A five-point evaluation framework (reconstruction fidelity, distribution matching, latent regularity, robust relevance, and computational tractability) provides systematic, model-agnostic criteria for assessing any neural network-based uncertainty set. We instantiate this framework with a Wasserstein Adversarial Autoencoder employing Gaussian mixture model-guided training for latent regularity and constraint-consistency regularisation for robust relevance. Restricting the decoder to ReLU activations enables exact worst-case verification through mixed-integer programming embedding. Extensive experiments on a production planning problem across six uncertainty distributions and six generative architectures, together with a multi-period facility location study, validate the framework and demonstrate that systematic attention to all five criteria yields uncertainty sets that are simultaneously expressive, well-calibrated, and optimisation-tractable.

Meta-reinforcement learning is a promising approach to multi-objective optimisation because it enables rapid policy adaptation across changing environments and preference settings. However, conventional few-shot methods usually fine-tune from a single shared meta-policy, which can reduce solution diversity and limit exploration of the Pareto front, especially in high-dimensional combinatorial problems such as supply chain optimisation. We propose a population-based Meta-reinforcement learning framework that combines decomposition with evolutionary search in scalarisation weight space. The framework maintains a population of weight vectors, each associated with a distinct meta-policy trained through gradient-based meta-learning, and iteratively refines this population through elitist selection, crossover, and mutation guided by hypervolume and entropy contributions. We evaluate the method in a multi-objective supply chain setting with conflicting economic, environmental, and social goals, and further test its generality on standard reinforcement learning problems. The results show that the proposed approach yields more diverse, better distributed Pareto front approximations, improves cross-task adaptation, increases hypervolume by up to 32% over Meta-multi-objective reinforcement learning in the complex case, and attains the lowest average Hausdorff distance among all compared methods.

The Jansen linkage is a single-degree-of-freedom planar leg mechanism whose eleven "holy numbers" were evolved by Theo Jansen to optimize the foot-path gait alone, with no regard for the wear of its revolute joints. This paper introduces a durability objective into the design of the Jansen leg. A parametric forward-kinematic model (two-circle-intersection solver), an inverse-dynamic model (constraint-Jacobian / Lagrange-multiplier formulation of a seven-body, ten-joint system, independently cross-verified by a reduced-DOF energy method), and an Archard wear model are coupled to evaluate, for any set of link lengths, both gait quality and the per-cycle sliding wear at every pin. Because the wear is computed on ideal, clearance-free revolute joints, the resulting wear figures are a relative comparative ranking rather than an absolute life prediction. A bi-objective problem -- composite gait error versus total joint wear, subject to step-length, ground-clearance, duty-factor and assembly constraints -- is solved with NSGA-II. Under the adopted gait metric the classical Jansen design is Pareto-dominated: for a representative design, link-length adjustments within +/-29% simultaneously flatten the stance (-28%), smooth the stance velocity (-58%) and reduce total joint wear by ~56%. A sensitivity study shows the wear advantage is robust across a crank-speed x payload envelope (48%-56%) and identifies the link lengths that most strongly govern wear. A variance-based global (Sobol) analysis confirms that two link lengths dominate the wear variance, and a Monte-Carlo manufacturing-tolerance study shows the wear advantage degrades gracefully under realistic fabrication error. The framework provides a practical route to longer-lived walking linkages and a baseline for future wear-clearance-impact coupled studies.

We present ENLIGHT, a fast and training free framework for low-light image enhancement based on direct optimization of a perceptual objective. Unlike deep learning approaches that require large scale training data and supervision, ENLIGHT operates in a zero-shot manner by optimizing image quality at inference time. The method employs a two stage global to local optimization strategy. In the first stage, ENLIGHT performs global illumination adjustment to improve visibility while maintaining structural consistency and avoiding excessive noise enhancement. In the second stage, a shadow aware refinement selectively improves low-intensity regions through masked local optimization, enhancing visibility without overexposure. To balance quality and efficiency, we introduce two modes: Fast, which uses a multi-objective formulation combining entropy, gradient preservation, and noise regularization, and Ultrafast, which reduces computational cost via a lightweight approximation of the same objective. The framework is optimizer agnostic and supports both evolutionary and lightweight local search methods. Experiments on BAID, Backlit300, LIME, MEF, NPE, and DICM demonstrate that ENLIGHT achieves competitive perceptual quality (MUSIQ, NIQE, BRISQUE) with significantly lower inference time. Qualitative results further show improved contrast, preserved structural details, and controlled noise amplification, making ENLIGHT a practical and interpretable alternative to learning based methods.

We introduce BASIL, a user-friendly desktop application for process optimization. BASIL employs a Bayesian approach, incorporating special acquisition functions that can be used to solve both single and multi-objective optimization problems. It provides a graphical interface that enables users to input their experimental parameters, optimization objectives, and legacy data. This is then used to build surrogate models, which are coupled with acquisition functions to guide and optimize a process towards a desired objective. To facilitate model building, BASIL provides a variety of predefined surrogate model templates. BASIL can be used to optimize any arbitrary experiment or process with known, user-defined input variables, optimization objectives, and defined output.